Subj. More detailed:
If I have JS-code on the page, loaded from my site, can that code make POST-request to the third site and set up cookies for that request?
Or more commonly – can I set up arbitrary headers (including cookies) in the JavaScript POST-request to the third-party site?
For security reasons that’s not possible
You can have a hidden iframe tag into your html, but you can’t access the cookies that the site saves into the browser